theorem 9.39 Floquet

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1373 · p. 288

Rests on

depends_on corollary 9.27 Linear stability ¶
- depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
  - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
    - depends_on definition 5.56 Endomorphism ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
  - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
    - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on definition 9.2 Linear equation; homogeneity ¶
      - depends_on remark 9.3 Normal form and first-order systems ¶
      - depends_on theorem 9.8 Picard–Lindelöf ¶
      - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
    - depends_on proposition 9.22 The exponential and its derivative ¶
      - depends_on definition 9.21 Matrix exponential ¶
      - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - depends_on theorem 7.51 Termwise differentiation ¶
      - proves proof ch:07-odes-sturm-liouville@proof-11 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-12 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-13 ¶
- depends_on theorem 9.26 Structure of the solutions ¶
  - depends_on theorem 8.24 Residue theorem ¶
    - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on definition 6.17 Simply connected space ¶
      - depends_on theorem 8.12 Cauchy ¶
      - proves proof ch:06-complex-analysis@proof-9 ¶
    - depends_on definition 8.22 Isolated singularities; residue ¶
      - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
      - depends_on theorem 8.21 Laurent expansion ¶
    - depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶ ↺
    - depends_on theorem 8.21 Laurent expansion ¶ ↺
    - proves proof ch:06-complex-analysis@proof-17 ¶
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - depends_on definition 5.70 Characteristic polynomial ¶
      - depends_on definition 5.56 Endomorphism ¶ ↺
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶
      - proves proof ch:03-linear-algebra-representations@proof-25 ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-18 ¶
    - proves proof ch:03-linear-algebra-representations@proof-26 ¶
  - depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
  - proves proof ch:07-odes-sturm-liouville@proof-14 ¶
- proves proof ch:07-odes-sturm-liouville@proof-15 ¶
depends_on definition 9.37 Fundamental matrix ¶
- depends_on corollary 9.9 Linear equations: existence on the whole interval ¶ ↺
- depends_on theorem 9.13 Dimension of the solution space ¶
  - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶ ↺
  - depends_on definition 9.2 Linear equation; homogeneity ¶ ↺
  - proves proof ch:07-odes-sturm-liouville@proof-6 ¶
depends_on proposition 9.38 Liouville's formula ¶
- depends_on definition 9.37 Fundamental matrix ¶ ↺
- depends_on proposition 9.15 Abel's identity ¶
  - depends_on definition 9.2 Linear equation; homogeneity ¶ ↺
  - depends_on definition 9.14 Wronskian ¶
  - proves proof ch:07-odes-sturm-liouville@proof-7 ¶
- proves proof ch:07-odes-sturm-liouville@proof-20 ¶
proves proof ch:07-odes-sturm-liouville@proof-21 ¶

Supports

depends_on corollary 9.41 Hill's equation ¶
- depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Linear stability	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1397
`depends_on`	→	Fundamental matrix	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1397
`depends_on`	→	Liouville's formula	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1397
`depends_on`	←	Hill's equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1491
`proves`	←	ch:07-odes-sturm-liouville@proof-21	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1401